We show that the generalized SQG equation on the plane is locally well-posed in spaces of low regularity solutions (essentially Hölder continuous with Hölder exponents depending on the equation parameter α (0, 12) ) that have H² level sets (i. e. , with L² curvatures). Moreover, for α 16 and initial data satisfying some additional hypotheses we show that the corresponding solutions can stop existing only when their level sets lose H²-regularity, and hence not just due to level set collisions or "pile ups".
Jeon et al. (Thu,) studied this question.